Problem: Simplify the following expression and state the conditions under which the simplification is valid. You can assume that $p \neq 0$. $q = \dfrac{7p + 49}{p - 10} \times \dfrac{p^2 - 7p - 30}{p^2 + 3p} $
Solution: First factor the quadratic. $q = \dfrac{7p + 49}{p - 10} \times \dfrac{(p - 10)(p + 3)}{p^2 + 3p} $ Then factor out any other terms. $q = \dfrac{7(p + 7)}{p - 10} \times \dfrac{(p - 10)(p + 3)}{p(p + 3)} $ Then multiply the two numerators and multiply the two denominators. $q = \dfrac{ 7(p + 7) \times (p - 10)(p + 3) } { (p - 10) \times p(p + 3) } $ $q = \dfrac{ 7(p + 7)(p - 10)(p + 3)}{ p(p - 10)(p + 3)} $ Notice that $(p + 3)$ and $(p - 10)$ appear in both the numerator and denominator so we can cancel them. $q = \dfrac{ 7(p + 7)\cancel{(p - 10)}(p + 3)}{ p\cancel{(p - 10)}(p + 3)} $ We are dividing by $p - 10$ , so $p - 10 \neq 0$ Therefore, $p \neq 10$ $q = \dfrac{ 7(p + 7)\cancel{(p - 10)}\cancel{(p + 3)}}{ p\cancel{(p - 10)}\cancel{(p + 3)}} $ We are dividing by $p + 3$ , so $p + 3 \neq 0$ Therefore, $p \neq -3$ $q = \dfrac{7(p + 7)}{p} ; \space p \neq 10 ; \space p \neq -3 $